The Curious Collatz Conjecture

Select any whole number greater than zero then perform the following operation:

If the number is even, divide it by 2.

If the number is odd, multiply it by 3 and add 1.

It’s said that if you continue to repeatedly apply the process, you’d always end up at 1. This is the premise of the Collatz conjecture, also known as the 3x + 1 Problem. The conjecture was named after German mathematician Lothar Collatz, who first suggested it in 1937.

Lothar Collatz

Until now, no one knows whether the conjecture is correct. As of 2017, the conjecture has been tested using computers for all numbers up to 87 × 260. The result indicated that the conjecture is valid for all the numbers checked. Even if it took several steps for some numbers, gradually, they all ended in (4, 2, 1) cycle.

Here are the numbers that take the longest to converge to 1 (For 101 to 1010):

less than 10 is 9, which has 19 steps,less than 100 is 97, which has 118 steps,less than 1,000 is 871, which has 178 steps,less than 10,000 is 6,171, which has 261 steps,less than 100,000 is 77,031, which has 350 steps,less than 1 million is 837,799, which has 524 steps,less than 10 million is 8,400,511, which has 685 steps,less than 100 million is 63,728,127, which has 949 steps,less than 1 billion is 670,617,279, which has 986 steps,less than 10 billion is 9,780,657,630, which has 1132 steps, andless than 100 billion is 75,128,138,247, which has 1228 steps.

The experiment may look convincing but we cannot conclude that the conjecture is true based on this. It only takes a counterexample to prove the conjecture wrong. This was what happened to the Mertens conjecture, Pólya conjecture, and Skewes’ number when they were debunked with a large counterexample.

But proving this innocent looking and seemingly simple conjecture is not simple at all. Many have tried to prove or disprove it but failed. When Paul Erdős learned of this problem, he remarked, “Mathematics is not yet ready for such confusing, troubling, and hard problems.” Erdős believed that we still do not possess the knowledge and tools powerful enough to solve this problem. He then offered $500 to anyone who can solve it

Paul Erdős

Nowadays, many mathematicians believe that the conjecture is true due to probabilistic heuristics. Nonetheless, until a proof of the validity of the Collatz conjecture is found, the conjecture will remain a conjecture.

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My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong.
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I wonder why such a conjecture was proposed. Just fun math brain work? (It seems that many mathematical studies are the work of intellectual curiosity rather than immediate applicability, with some discoveries finding application years after invention.)
Your assertion at the end that the Collatz Conjecture remains a conjecture, even with extensive computer analysis, reminds me of an oft-repeated line from a middle school science textbook: “Science is tentative!”

G. H. Hardy insisted in his book “A Mathematician’s Apology” that his works on pure mathematics don’t have real world applications in mind and he considered doing pure mathematics for the sake of finding real world application as silly.

Ironically, some of his works in number theory are now applied in cyber security.
It’s tentative until it’s proven. Many considered Fermat’s Last Theorem as valid long before it was proven by Wiles and they were correct. However, this is not always the case as many conjectures were destroyed by a very large ciunterexample before.

As for science being tentative, it’s so true. I can’t begin to imagine how people a thousand years later would react to our “science”. Many scientists these days thought themselves as gods or at least almost infallable. They would dismiss other theories that they don’t understand or that clash with their worldview, just like the ancients. Perhap, people of the future will see our follies and do something about it 🙂

I’ve heard that before, about pure mathematics existing for mathematics’ sake. I think pure mathematics is one of those fields that you either like or you don’t. I for, one, enjoy it, while my engineering-minded sister prefers applied, “practical” mathematics. I tracked down A History of Mathematics: From Babylonia to Modernity, and the first chapter on Babylonian mathematics addressed a similar point about mathematics not existing solely for real world application, though further investigation might find one. Scholars of the time would demonstrate mathematical finesse for mental exercise rather than exploration of application.

I read somewhere that mathematics researchers vying for sponsorship for their “useless” study sometimes point to past research, like number theory, to prove that what might seem pointless now may have grand uses in the future. According to Hardy, I guess that wouldn’t matter much.

Every generation thinks they have it set and right, until the next comes and enlightens them – or at least tries to enlighten them. Look how many years it took for the heliocentricity of the universe to displace the ancient, traditional view of geocentricity, even with scientific evidence.

As long as it can be proven indisputably (there are many types of mathematical proofs), the conjecture becomes a theorem.

The danger of assuming that the conjecture is true just because it’s valid on very large numbers is that somewhere along the line, there may be a much larger number that will prove the conjecture false.

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Learn Fun Facts is an archive of curious facts for the curious. I write on a variety of subjects, including mathematics, science, technology, language, history, literature, art and anything in between.